Consider a particle in a 1D half-infinite well. The potential in region III is infinite and...

Transcribed Image Text:

Consider a particle in a 1D half-infinite well. The potential in region III is infinite and is zero in region I. In this case however, the potential is not infinite in region II but has a constant value Vo. V(A)-{ 00 0 x<0 0sxsL x> L III 20 I X-L II Region III a. Write the Schrodinger equation in region III. What is the only acceptable solution? Region I b. Write the Schrodinger equation in region I. Show that , Asin(kx) + Bcos (kx) is solution of this equation and give the constant kas a function of the total energy E of the particle. c. Using continuity of the wave function at the boundary between region III and I, determine the constant B. Region II -ix d. Write the Schrodinger equation in region II. Show that ,,Ce + De is solution of this equation and give the constant à as a function of the total energy E<V, of the particle. e. Give the only acceptable value of the constant C. Justify. Continuity of the wavefunction f. Give the continuity conditions at x-L and infer an equation relating the parameters k, L and A. The constants A, B, C, and D should not appear in your final result. g. Demonstrate that the previous equation can be rearanged into the continuity equation: 2mV L² z tan ¹(z) = -√√(a²-z²) with z=kL_ and a= ħ² Consider a particle in a 1D half-infinite well. The potential in region III is infinite and is zero in region I. In this case however, the potential is not infinite in region II but has a constant value Vo. V(A)-{ 00 0 x<0 0sxsL x> L III 20 I X-L II Region III a. Write the Schrodinger equation in region III. What is the only acceptable solution? Region I b. Write the Schrodinger equation in region I. Show that , Asin(kx) + Bcos (kx) is solution of this equation and give the constant kas a function of the total energy E of the particle. c. Using continuity of the wave function at the boundary between region III and I, determine the constant B. Region II -ix d. Write the Schrodinger equation in region II. Show that ,,Ce + De is solution of this equation and give the constant à as a function of the total energy E<V, of the particle. e. Give the only acceptable value of the constant C. Justify. Continuity of the wavefunction f. Give the continuity conditions at x-L and infer an equation relating the parameters k, L and A. The constants A, B, C, and D should not appear in your final result. g. Demonstrate that the previous equation can be rearanged into the continuity equation: 2mV L² z tan ¹(z) = -√√(a²-z²) with z=kL_ and a= ħ²

Answer rating: 100% (QA)

a In region III x 0 the Schrodinger equation is 22m d2IIIdx2 III EIII The on... View the full answer